What is the probability density function (PDF) of the exponential distribution?

What is the probability density function (PDF) of the exponential distribution?

• A. f(x) = λe^(-λx)
• B. f(x) = (1/θ)e^(-x/θ)
• C. f(x) = θe^(-θx)
• D. f(x) = θe^(-x/θ)

Answer: (B)

The probability density function (PDF) of the exponential distribution describes the likelihood of a continuous random variable taking on a particular value, and it is characterized by a unique property: it is memoryless. The two common forms of the exponential distribution's PDF are often expressed in terms of the rate parameter, λ, or the scale parameter, θ. When using the rate parameter, λ, the PDF is given by f(x) = λe^(-λx) for x ≥ 0, which emphasizes how the rate at which events occur influences the distribution. In the other representation with the scale parameter, θ, typically used in contexts where the average rate of events is a primary focus, the PDF can be expressed as f(x) = (1/θ)e^(-x/θ) for x ≥ 0. This form is equivalent to the standard parameterization, where θ is the mean of the distribution. The expression presented in option B fits this standard formulation correctly, clearly indicating the relationship between the mean and the rate of occurrence, with the term (1/θ) denoting the rate of decline of the exponential function relative to θ. Thus, the PDF of the exponential distribution correctly captures the decay rate of probabilities and is vital in

The probability density function (PDF) of the exponential distribution describes the likelihood of a continuous random variable taking on a particular value, and it is characterized by a unique property: it is memoryless. The two common forms of the exponential distribution's PDF are often expressed in terms of the rate parameter, λ, or the scale parameter, θ.

When using the rate parameter, λ, the PDF is given by f(x) = λe^(-λx) for x ≥ 0, which emphasizes how the rate at which events occur influences the distribution. In the other representation with the scale parameter, θ, typically used in contexts where the average rate of events is a primary focus, the PDF can be expressed as f(x) = (1/θ)e^(-x/θ) for x ≥ 0. This form is equivalent to the standard parameterization, where θ is the mean of the distribution.

The expression presented in option B fits this standard formulation correctly, clearly indicating the relationship between the mean and the rate of occurrence, with the term (1/θ) denoting the rate of decline of the exponential function relative to θ.

Thus, the PDF of the exponential distribution correctly captures the decay rate of probabilities and is vital in